Case Study Questions for Class 9 on Number system

This page provides case study-based questions for Class 9 students focused on the Number System. Includes solved MCQs, theory explanations, and practice questions aligned with CBSE guidelines for effective exam preparation.

Grade 9 Number Systems: Case-Based Questions

Here are three case-based questions for Grade 9 students on Number Systems, along with theory, MCQs, and solutions:

Case Study 1: Representing Irrational Numbers on the Number Line

Theory:

Irrational numbers like √2, √3, and √5 cannot be expressed as exact fractions and have non-terminating, non-repeating decimal expansions. To represent √2 on the number line, we use the Pythagorean theorem. Construct a right-angled triangle with legs of 1 unit each; the hypotenuse will be √2. Using a compass, we transfer this length to the number line. Similarly, √3 is constructed using √2 as a base, forming a spiral of right triangles. This geometric method ensures accurate placement of irrational numbers.

Question:

A student is constructing √5 on the number line using the following steps:

  1. Draw a number line and mark 0 as point O.
  2. From O, measure 2 units to point A.
  3. At A, draw a perpendicular line and mark 1 unit upwards to point B.
  4. Join O to B.

Based on this, answer the following MCQs:

1. What is the length of OB?

  • a) 1
  • b) √3
  • c) √5 (Correct)
  • d) 3

2. How would you construct √3 using a similar method?

  • a) Start with 1 unit, perpendicular 1 unit
  • b) Start with 1 unit, perpendicular √2 units
  • c) Start with 1 unit, perpendicular 2 units
  • d) Start with 2 units, perpendicular 1 unit (Correct)

3. Which irrational number is constructed using legs 1 and √2?

  • a) √3
  • b) √5
  • c) √6 (Correct)
  • d) √7

4. What is the decimal approximation of √5?

  • a) 2.236 (Correct)
  • b) 2.414
  • c) 2.645
  • d) 2.828

5. Which theorem is used to justify this construction?

  • a) Thales’ theorem
  • b) Pythagoras theorem (Correct)
  • c) Euclid’s division lemma
  • d) Fundamental theorem of arithmetic

Solution:

  • 1. (c) √5 since OB = √(2² + 1²) = √5
  • 2. (d) For √3, use legs 2 and 1
  • 3. (c) √6 from legs √2 and 2
  • 4. (a) √5 ≈ 2.236
  • 5. (b) Pythagoras theorem

Case Study 2: Rationalization of Denominators

Theory:

Rationalization is the process of eliminating irrational numbers from the denominator of a fraction. For example, 1/√2 is rationalized by multiplying numerator and denominator by √2, yielding √2/2. This simplifies calculations and is essential for exact arithmetic. The key identity used is (√a × √a = a). For denominators like (1 + √3), we multiply by the conjugate (1 – √3) to use the difference of squares formula (a+b)(a-b) = a² – b².

Question:

A fraction 5/(2 + √3) needs to be rationalized. The steps are:

  1. Multiply numerator and denominator by the conjugate (2 – √3)
  2. Apply the identity (a+b)(a-b) = a² – b²

MCQs:

1. What is the conjugate of 2 + √3?

  • a) 2 – √3 (Correct)
  • b) -2 + √3
  • c) √3 – 2
  • d) 2 + √3

2. After rationalization, the denominator becomes:

  • a) 1
  • b) 4 – 3 = 1 (Correct)
  • c) 4 + 3 = 7
  • d) 2 – √3

3. The rationalized form of 5/(2 + √3) is:

  • a) 10 – 5√3 (Correct)
  • b) 10 + 5√3
  • c) 5(2 – √3)/7
  • d) 5(2 – √3)

4. Which identity is used in rationalization?

  • a) (a+b)² = a² + b² + 2ab
  • b) (a-b)² = a² + b² – 2ab
  • c) (a+b)(a-b) = a² – b² (Correct)
  • d) a² – b² = (a-b)²

5. What is the rationalized form of 1/√5?

  • a) √5
  • b) √5/5 (Correct)
  • c) 5/√5
  • d) 1/5

Solution:

  • 1. (a) Conjugate is 2 – √3
  • 2. (b) 2² – (√3)² = 4 – 3 = 1
  • 3. (a) 5(2 – √3) = 10 – 5√3
  • 4. (c) Difference of squares identity is used
  • 5. (b) √5/5 is the rationalized form

Case Study 3: Laws of Exponents for Real Numbers

Theory:

The laws of exponents for real numbers are:

  1. am × an = am+n
  2. am/an = am-n
  3. (am)n = amn
  4. a-m = 1/am
  5. a0 = 1 for a ≠ 0

These laws apply to all real numbers, including irrational exponents. For example, 2√2 × 2√3 = 2√2 + √3. Understanding these rules is crucial for simplifying expressions and solving equations.

Question:

Simplify the expression [(3√5)2 × 3-2]/[32√5 – 2].

MCQs:

1. What is (3√5)2 simplified?

  • a) 32√5 (Correct)
  • b) 3√10
  • c) 9√5
  • d) 35

2. The expression [32√5 × 3-2]/[32√5 – 2] simplifies to:

  • a) 1 (Correct)
  • b) 34√5 – 4
  • c) 3-4
  • d) 32√5 + 2

3. Which law is used in (am)n = amn?

  • a) Power of a power (Correct)
  • b) Product of powers
  • c) Quotient of powers
  • d) Negative exponent

4. If 2x = 8√2, what is x?

  • a) 3√2 (Correct)
  • b) 2√3
  • c) 4√2
  • d) 6

5. What is 50 + (2-1)-2?

  • a) 5
  • b) 4
  • c) 1 + 4 = 5 (Correct)
  • d) 1 + 1/4 = 5/4

Solution:

  • 1. (a) 32√5
  • 2. (a) Numerator and denominator simplify to same power, result is 1
  • 3. (a) Power of a power law
  • 4. (a) 8 = 2³, so 8√2 = (2³)√2 = 23√2
  • 5. (c) 50 = 1, (2-1)-2 = 2² = 4, sum is 5